Quantum adiabatic machine learning


We develop an approach to machine learning and anomaly detection via quantum adiabatic evolution. This approach consists of two quantum phases, with some amount of classical preprocessing to set up the quantum problems. In the training phase we identify an optimal set of weak classifiers, to form a single strong classifier. In the testing phase we adiabatically evolve one or more strong classifiers on a superposition of inputs in order to find certain anomalous elements in the classification space. Both the training and testing phases are executed via quantum adiabatic evolution. All quantum processing is strictly limited to two-qubit interactions so as to ensure physical feasibility. We apply and illustrate this approach in detail to the problem of software verification and validation, with a specific example of the learning phase applied to a problem of interest in flight control systems. Beyond this example, the algorithm can be used to attack a broad class of anomaly detection problems.

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  1. 1.

    Random number generation may appear to be a counterexample, as it is multi-valued, but only over different calls to the random-number generator.

  2. 2.

    One important consideration is that, as we shall see below, for practical reasons we may only be able to track errors at the level of one-bit errors and correlations between bit-pairs. Such limited tracking can be alleviated to some extent by using intermediate spaces, where higher order correlations between bits appearing at the level of the output space may not yet have had time to develop.

  3. 3.

    This inequality reflects the fact that for \(n\) overlapping sets, \(P\left[\bigcup _{i=1}^{n}s_i\right]=\sum _{i=1}^{n}P[s_i]-\sum _{i\ne j}P[s_i\cap s_j] + \sum _{i\ne j \ne k}P[s_i\cap s_j \cap s_k] - \sum _{i\ne j\ne k \ne m}P[s_i\cap s_j \cap s_k \cap s_m]+\dots \) Each term is larger than the next in the series; \(n+1\) sets cannot intersect where \(n\) sets do not. Our truncation of the series is greater than or equal to the full value because we stop after a subtracted term.

  4. 4.

    Any Boolean function of \(\ell \) variables can be uniquely expanded in the form \(f_i(x_1,\dots ,x_\ell ) = \sum _{\alpha =0}^{2^\ell -1} \epsilon _{i\alpha } s_{\alpha }\), where \(\epsilon _{i\alpha }\in \{0,1\}\) and \(s_\alpha \) are the \(2^\ell \) “simple” Boolean functions \(s_0 = x_1 x_2 \cdots x_\ell \), \(s_1 = x_1 x_2 \cdots \overline{x_\ell }\), \(\dots \), \(s_{2^\ell -1} = \overline{x_1}\, \overline{x_2} \cdots \overline{x_\ell }\), where \(\overline{x}\) denotes the negation of the bit \(x\). Since each \(\epsilon _{i\alpha }\) can assume one of two values, there are \(2^{2^\ell }\) different Boolean functions.

  5. 5.

    We are grateful to Greg Tallant from the Lockheed Martin Corporation for providing us with this problem as an example of interest in flight control systems.


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The authors are grateful to the Lockheed Martin Corporation for financial support under the URI program. KP is also supported by the NSF under a graduate research fellowship. DAL acknowledges support from the NASA Ames Research Center.

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Correspondence to Kristen L. Pudenz.

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Pudenz, K.L., Lidar, D.A. Quantum adiabatic machine learning. Quantum Inf Process 12, 2027–2070 (2013). https://doi.org/10.1007/s11128-012-0506-4

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  • Adiabatic quantum computation
  • Quantum algorithms
  • Software verification and validation
  • Anomaly detection